Motivated by the Klebanov-Polyakov conjecture we investigate the O(N) vector model at large N on a squashed three-sphere and its holographic relation to bulk gravity on asymptotically locally AdS4 space. We present analytical results for the action of the field theory as the squashing parameter , alpha→ -1, when the boundary becomes effectively one dimensional. In this limit we solve the theory exactly and show that the action of the boundary theory scales as ln(1 + alpha)/(1 + alpha)[2] which is to be compared and contrasted with the -1/(1 + alpha)[2] scaling of gravity in AdS-Taub-NUT space. These results are consistent with the numerical evidence presented in hep- th/0503238, and the soft logarithmic departure is interpreted as a prediction for the contribution due to higher spin fields in the bulk AdS[4] geometry. We then give an introduction to non relativistic AdS/CFT and numerically compute the inter quark potential in the non relativistic theory obtained by taking the DLCQ of N = 4 SYM theory by considering Wilson loops in the dual string theory via the AdS/CFT correspondence.

The main theme of this thesis is the formulation of self-duality for extended objects (p-branes). An approach to self-duality for membranes is developed using the correspondence between the large N limit of SU(N) gauge theories and the membrane theory. This correspondence is established via the use of the coadjoint orbit method. It is shown that classical gauge field theories can be formulated on the coadjoint orbits of an infinite dimensional group (a semidirect product of the group of gauge transformations and the Heisenberg-Weyl group); in Chap. II this construction is carried out for Yang-Mills, Cherns-Simons, topological Yang-Mills and F A B theories, as well as the Wess-Zumino-Novikov-Witten model. In Chap. III it is shown that for homogeneous fields (i.e. gauge mechanics) and in the N -1-" limit, the coadjoint orbit action becomes identical to the membrane action in the light cone gauge. The self-duality equations for gauge fields then translate into the self-duality equations for membranes. In Chap. IV another approach is developed, one which allows us to formulate the self-duality equations for a much larger class of extended objects. This generalized self-duality is based on the notion of p-fold vector products. We exhibit several classes of solutions for these generalized self-dual extended objects and classify all the cases in which they exist. We also show that the self-intersecting string instantons, introduced by Polyakov constitute a special case of these solutions. Of particular interest are two octonionic classes: a membrane in 7 dimensions and a 3-brane in 8 dimensions. To simplify the calculations in these cases we developed an approach to octonionic symbolic computing making use of "Mathematica". Some possible applications of self-dual extended objects are briefly discussed. / Ph. D.

published_or_final_version / Physics / Doctoral / Doctor of Philosophy

In this thesis we explore certain limits of the AdS/CFT correspondence for integrability. This is done by calculating the action of the dilatation operator on operators known as restricted Schur polynomials, which are AdS/CFT dual to D3-branes known as giant gravitons. We focus on operators in N = 4 super-Yang-Mills theory, which is dual to type IIB string theory on an AdS5×S5 background. We find that, in various cases, this theory is integrable in a large N non-planar limit.

Recently it has been shown that N = 4 super Yang-Mills theory is integrable in the planar limit. Past arguments suggest the integrability is only present in the planar limit. However, this conclusion was shown to be incorrect. Two speci c classes of operators were studied in the O(N) sector. The rst were labelled by Young diagrams having two long columns. The second were labelled by Young diagrams having two long rows. This result was then generalized to p long rows or columns with p xed to be O(1) as N ! 1. For this case, the non-planar limit was found to be integrable. In this dissertation, we extend this work by considering p to be O(N). We have calculated the dilation operator for the case with two impurities.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. March 2013. / In the past decade, the gauge/gravity duality has been extensively explored in the large N limit. In particular, the spectrum of anomalous dimensions have been compared with the energy spectrum of the dual string theory showing remarkable agreement. In this limit, for operators with a bare dimension of order 1, planar diagrams give the leading contribution to the anomalous dimension. To obtain the anomalous dimensions, one needs to diagonalize the dilatation operator. One of the methods used to accomplish this is integrability. This allows an exact computation of the spectrum of the anomalous dimensions. There is by now a great deal of evidence that N = 4 supersymmetric Yang-Mills (SYM) theory and N = 6 superconformal Chern Simons (ABJ(M)) theory are integrable in the planar limit. In this thesis we probe the gauge gravity duality at finite N using novel tools developed from the representation theory of symmetric and unitary groups. We start by studying the action of the nonplanar dilatation operator of N = 4 SYM theory and ABJ(M) theory. The gauge invariant operators we consider are the restricted Schur polynomials. In the case of N = 4 SYM theory, we obtain the spectrum of the anomalous dimension beyond the SU(2) sector at one loop, and in the SU(2) sector at two loops. In both cases, we obtain the spectrum at arbitrary (finite) N. We then obtain the spectrum of anomalous dimensions in the SU(2) sector of ABJ(M) theory at two loops. The class of gauge invariant operators we consider have classical dimension of order O(N). In both theories, the spectrum of the anomalous dimensions reduces to a set of decoupled harmonic oscillators at large N. This indicates, for the first time, that N = 4 SYM theory and ABJ(M) theory exhibit nonplanar integrability. We expect to recover non-perturbative quantum gravity effects, from the gauge/gravity duality, when N is finite. The non-planar integrability we discover here may play an important role in finite N studies of the gauge/gravity duality, and hence may play an important role in understanding non-perturbative string stringy physics. In addition, we study various classes of correlators in ABJ(M) theory. In this context, we derive extremal n-point correlators in ABJ(M) theory and we probe the giant graviton dynamics in these theories.

A research project submitted to the Faculty of Science, University of the Witwatersrand, in fulfillment for the degree of Master of Science in Physics. May 25, 2017. / This project addresses the study of the memory effect. We review the effect in electromagnetism, which is an abelian gauge theory. We prove that we can shift the phase factor by performing a gauge transformation. The gauge group is U(1). We extend the study to the nonabelian gauge theory by computing the memory in SU(2) which vanishes up to the first order Taylor expansion. Keywords: Memory Effect, Aharonov-Bohm effect, Nonabelian Gauge Theory, Supersymmetry / GR2018

No description available.

In this thesis, we study two problems focusing on the interplay between geometric properties of differential equations and their invariants. / For the first project, we study the validity of the principle of symmetric criticality (PSC) in the context of invariant gauge fields over the four-dimensional non-reductive pseudo-Riemannian homogeneous spaces G/K recently classified by Fels & Renner (2006). Given H compact semi-simple, classification results are obtained for principal H-bundles over G/K admitting: (1) a G-action (by bundle automorphisms) projecting to left multiplication on the base, and (2) at least one G-invariant connection. There are two cases which admit nontrivial examples of such bundles and all G-invariant connections on these bundles are Yang--Mills. Using the invariant criteria obtained by Anderson--Fels--Torre, the validity of PSC is investigated for the bundle of connections and is shown to fail for all but one of the Fels--Renner cases. This failure arises from degeneracy of the scalar product on pseudo-tensorial forms restricted to the space of symmetric variations of an invariant connection. In the exceptional case where PSC is valid, there is a unique G-invariant connection which is moreover universal, i.e. it is a solution of the Euler--Lagrange equations associated to any G-invariant Lagrangian on the bundle of connections. This solution is a canonical connection associated with a weaker notion of reductivity which we introduce. / The second project is a study of the contact geometry of scalar second order hyperbolic equations in the plane of generic type. Following a derivation of parametrized contact-invariants to distinguish Monge--Ampere (class 6-6), Goursat (class 6-7) and generic (class 7-7) hyperbolic equations, we use Cartan's equivalence method to study the generic case. An intriguing feature of this class of equations is that every generic hyperbolic equation admits at most a nine-dimensional contact symmetry algebra. The nine-dimensional bound is sharp: normal forms for the contact-equivalence classes of these maximally symmetric equations are derived and explicit symmetry algebras are presented. Moreover, all such equations are Darboux integrable. An enumeration of several submaximally symmetric (eight and seven-dimensional) structures is also given.

No description available.